MinT - Best Known (117, 117+133, s)-Nets in Base 3

Best Known (117, 117+133, s)-Nets in Base 3

(117, 117+133, 74)-Net over F₃ — Constructive and digital

Digital (117, 250, 74)-net over F₃, using

t-expansion [i] based on digital (107, 250, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(117, 117+133, 120)-Net over F₃ — Digital

Digital (117, 250, 120)-net over F₃, using

t-expansion [i] based on digital (113, 250, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(117, 117+133, 738)-Net in Base 3 — Upper bound on s

There is no (117, 250, 739)-net in base 3, because

1 times m-reduction [i] would yield (117, 249, 739)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66182 858515 648513 366851 464896 177431 443635 946274 375105 844575 637177 852312 436445 152231 985606 327856 160499 321022 835074 117989 > 3²⁴⁹ [i]